Topic: Control Systems
A sensitivity of transfer function given by $T=(A_1+kA_2)/(A_3+kA_4)$ with respect to parameter k is
- A. $k(A_2A_3-A_1A_4)/((A_3+kA_4)(A_1+kA_2))$
- B. $(A_2A_3-A_1A_4)/((A_3+kA_4)^2)$
- C. $k(A_2A_3-A_1A_4)/((A_3+kA_4)^2)$
- D. $(A_2A_3-A_1A_4)/((A_3+kA_4)(A_1+kA_2))$
Answer: A. $k(A_2A_3-A_1A_4)/((A_3+kA_4)(A_1+kA_2))$
Explanation: The sensitivity is defined as $S_k^T = \frac{\partial T}{\partial k} \frac{k}{T}$. Differentiating $T$ using the quotient rule yields $\frac{\partial T}{\partial k} = \frac{A_2(A_3+kA_4) - A_4(A_1+kA_2)}{(A_3+kA_4)^2} = \frac{A_2A_3 - A_1A_4}{(A_3+kA_4)^2}$. Multiplying by $k/T = \frac{k(A_3+kA_4)}{A_1+kA_2}$ gives $S_k^T = \frac{k(A_2A_3 - A_1A_4)}{(A_3+kA_4)(A_1+kA_2)}$.